| A model for rapid directional solidification is considered that has a velocity-dependent distribution coefficient and liquidus slope, and a linear form for attachment kinetics. The linear stability analysis of Merchant and Davis revealed that in addition to the steady Mullins and Sekerka cellular mode there is a zero wavenumber pulsatile mode that may be most dangerous in the linear theory.;Two strongly-nonlinear analyses are performed that describe the pulsatile mode only. In the first (second) case, non-equilibrium effects that alter solute rejection at the interface are taken asymptotically small (large). A nonlinear-oscillator equation governs the position of the solid-liquid interface at leading order, and amplitude- and phase-evolution equations are derived for the uniformly-pulstating interface. The results are used to make predictions about the characteristics of solute bands that would be frozen into the solid.;We perform a weakly-nonlinear analysis near the onset of the steady cellular mode. The transition to cells is smooth (supercritical) for high speeds and a jump (sub-critical) for low speeds. The steady branch may terminate at high speeds, however, if an oscillatory mode is present.;A weakly-nonlinear analysis is performed in the neighborhood of the codimension-2 point where the onset of the oscillatory mode and the cellular mode coalesce. A coupled set of evolution equations is derived that governs the slowly-varying amplitudes of the pulsations and cells, and the bifurcation structures are determined.;A weakly-nonlinear analysis of the oscillatory mode near its onset yields a complex Ginzburg-Landau equation governing the evolution of the solid-liquid interface shape. Parametric regions demarking bifurcation type are presented, and spatial pattern formation is discussed in two and three dimensions.;Finally, we consider energy stability theory for the directional solidification of a dilute binary alloy under local equilibrium conditions at the interface. Energy theory is used to predict the conditions necessary for stability of the planar interface against finite-amplitude disturbances. |
